Theoretical Prediction of Structural, Mechanical, and Thermophysical Properties of the Precipitates in 2xxx Series Aluminum Alloy

: We presented a theoretical study for the structural, mechanical, and thermophysical properties of the precipitates in 2xxx series aluminum alloy by applying the widely used density functional theory of Perdew-Burke-Ernzerhof (PBE). The results indicated that the most thermodynamically stable structure refers to the Al 3 Zr phase in regardless of its different polymorphs, while the formation enthalpy of Al 5 Cu 2 Mg 8 Si 6 is only -0.02 eV (close to zero) indicating its metastable nature. The universal anisotropy index of A U follows the trend of: Al 2 Cu > Al 2 CuMg ≈ Al 3 Zr_D0 22 ≈ Al 20 Cu 2 Mn 3 > Al 3 Fe ≈ Al 6 Mn > Al 3 Zr_D0 23 ≈ Al 3 Zr_L1 2 > Al 7 Cu 2 Fe > Al 3 Fe 2 Si. The thermal expansion coef-ﬁcients (TECs) were calculated based on Quasi harmonic approximation (QHA); Al 2 CuMg shows the highest linear thermal expansion coefﬁcient (LTEC), followed by Al 3 Fe, Al 2 Cu, Al 3 Zr_L1 2 and others, while Al 3 Zr_D0 22 is the lowest one. The calculated data of three Al 3 Zr polymorphs follow the order of L1 2 > D0 23 > D0 22 , all of them show much lower LTEC than Al substance. For multi-phase aluminum alloys, when the expansion coefﬁcient of the precipitates is quite different from the matrix, it may cause a relatively large internal stress, or even produce cracks under actual service conditions. Therefore, it is necessary to discuss the heat misﬁt degree during the material design. The discrepancy between a -Al and Al 2 CuMg is the smallest, which may decrease the heat misﬁt degree between them and improve the thermal shock resistant behaviors.


Introduction
2xxx series aluminum alloys are widely used in spacecraft parts, engine pistons, aircraft structures, missile components, propellers, aircraft skeletons, etc.The 2xxx series aluminum alloys mainly include Al-Cu, Al-Cu-Mg, Al-Cu-Mg-Mn-Zr alloys, etc., which could be strengthened by heat treatment, and thereby the hardness and strength will be further improved after solutionizing treatment and the following artificial aging treatment [1][2][3].The 2xxx series aluminum alloy, in particular, keeps good performance at high temperatures and is commonly used in weldable forgings and structural parts.
During artificial aging treatment of 2xxx series aluminum alloys, etc., a succession of precipitates is developed from the supersaturated solid solution of α-Al.θ-Al 2 Cu, is obtained from the supersaturated solid solution, and then goes through the GP zone (Guinier-Preston zone), then the metastable θ", θ' and finally the θ-Al 2 Cu phase; S-Al 2 CuMg undergoes a similar process to θ-Al 2 Cu [2].Zr and Mn are also frequently used to improve the strength of the alloys by forming intermetallics such as Al 3 Zr, Al 20 Cu 2 Mn 3 , and Al 6 Mn.Particularly, Si and Fe elements are impurities in aluminum alloys, which exist as Al 3 Fe, Al 3 Fe 2 Si, Al 5 Cu 2 Mg 8 Si 6 , and Al 7 Cu 2 Fe.
For θ-Al 2 Cu and S-Al 2 CuMg phases, a lot of experimental work has been conducted to reveal their microstructures, mechanical and physical properties.Kairy et al. [4] claimed the hardness of the 2xxx series alloy can be increased with Sc and Zr additions.The fine spherical Al 3 (ScZr) effectively retards the recrystallization process, which benefits the high-temperature mechanical property of aluminum alloys.Actually, the ground state of Al 3 Zr has a tetragonal D0 23 structure, while the L1 2 structure can be stabilized by Cu or Mn [5].Al 20 Cu 2 Mn 3 is a kind of common dispersoid in 2xxx series aluminum alloys, which is formed during the homogenization [6].
The high temperature mechanical strength of aluminum alloys can be improved by fine and uniformly distributed Al 20 Cu 2 Mn 3 particles [7].Al 20 Cu 2 Mn 3 has an orthorhombic structure with the space group of Bbmm [8].Shen et al. [9] reported the atomic arrangement of the Al 20 Cu 2 Mn 3 structure; and the optimized lattice parameters were estimated as a = 23.98Å, b = 12.54 Å, and c = 7.66 Å. Huang et al. [10] reported Al 6 Mn is less steady than Al 3 Fe or Al 3 Fe 2 Si from the energetic point of view.Zhu et al. [11] found a high-strength die-cast aluminum alloy by optimizing the synergistic strengthening of Q-Al 5 Cu 2 Mg 8 Si 6 and θ-Al 2 Cu phases, the yield strength of 225 MPa and elongation of 4.3% were obtained.For the Al 7 Cu 2 Fe phase, Tian et al. [12] reported the bulk, shear and young's moduli are 107.8,74.5, and 181.7 GPa, respectively.
Although many investigations associated with precipitates have been performed in previous experimental and theoretical work, most of them are mainly focused on the structures and mechanical strength.To date, little information is available about the thermodynamical stability, anisotropic mechanical property, temperature dependence thermal expansion, and thermal capacity due to the difficulties of experiments.Based on the importance of precipitates in 2xxx series aluminum alloys, the properties mentioned above should be investigated and discussed to reveal the intrinsic behavior of the 2xxx series aluminum alloys.In this paper, we will perform a comprehensive study on these properties of the precipitates using first-principles calculations based on the density functional theory (DFT).

Model and Computational Method
The crystal structures of precipitates considered in current paper are shown in Figure 1, which include Al 2 Cu, Al 2 CuMg, Al 3 Fe, Al 3 Fe 2 Si, Al 5 Cu 2 Mg 8 Si 6 , Al 7 Cu 2 Fe, Al 6 Mn, Al 20 Cu 2 Mn 3 , and Al 3 Zr.All these phases are built by their conventional crystal state, and the crystal structures and lattice parameters are shown in Table 1.This work was carried out by using the Cambridge Sequential Total Energy Package (CASTEP) code based on density functional theory (DFT) [13,14].The criteria for convergence were 10 −8 eV/atom for total energy, and 10 −4 eV/Å for Hellmann-Feynman forces, respectively.The Broyden-Fletcher-Goldfarb-Shannon (BFGS) algorithm is applied to optimize the crystal structure including lattice parameters and atomic fractional coordinates.The ultrasoft pseudo-potentials (USPPs) were used to represent the interactions between the ionic core and valence electrons.A plane-wave basis set with E cut of 500 eV was used.The exchange and correlation relationship of Perdew-Burke-Ernzerhof (PBE) was applied for calculations [15].For kspace, summation the 0.3 Å −1 for all phases with Monkhorst-Pack scheme in the first irreducible Brillouin zone [16] has been used.The valence electrons were considered as 3s 2 3p 1 , 3d 10 4s 1 , 2p 6 3s 2 , 3s 2 3p 2 , 3d 5 4s 2 , and 4s 2 4p 6 4d 2 5s 2 for Al, Cu, Mg, Si, Mn, and Zr, respectively.In order to study the mechanical response of the crystals to external stress, the elastic properties are determined using the stress-strain relationship by deforming the unit cell using lagrangian strain modes based on Hooker's law [17].
In order to calculate the thermal expansion coefficient (TEC), the Helmholtz free energy was given by F (V, T) = E gs (V) + F vib (V, T) + F ele (V, T) [18].The E gs refers to total energy with ground-state obtained directly by DFT calculation at 0 K.The vibrational free energy (F vib ) was calculated by means of the quasi-harmonic approximation (QHA) based on the empirical Debye model [19,20].F ele is the electron thermal excitations at finite temperature and can be calculated by Mermin statistics F ele = E ele − TS ele .Using isothermal curves (F (V, T) − V), the equilibrium volumes (V) at different temperatures can be obtained from the Birch-Murnaghan equation of state (EOS) [21].Finally, the volumetric TEC λ (T) can be determined by λ = 1 where E(V) is the total energy, B 0 , V 0 , and E 0 refer to the equilibrium bulk modulus, volume and energy, and B 0 is the pressure derivation.In order to calculate the thermal expansion coefficient (TEC), the Helmholtz free energy was given by F (V, T) = Egs(V) + Fvib (V, T) + Fele (V, T) [18].The Egs refers to total energy with ground-state obtained directly by DFT calculation at 0 K.The vibrational free energy (Fvib) was calculated by means of the quasi-harmonic approximation (QHA) based on the empirical Debye model [19,20].Fele is the electron thermal excitations at finite temperature and can be calculated by Mermin statistics Fele = Eele − TSele.Using isothermal curves (F (V, T) − V), the equilibrium volumes (V) at different temperatures can be obtained from the Birch-Murnaghan equation of state (EOS) [21].Finally, the volumetric TEC λ (T) can be determined by where E(V) is the total energy, B0, V0, and E0 refer to the equilibrium bulk modulus, volume and energy, and B0 ′ is the pressure derivation.

Mechanical Properties
Table 1 listed the equilibrium lattice parameters, formation enthalpies and cohesive energy of precipitates in 2xxx series aluminum alloys.The optimized lattice parameters are in good agreement with the references.The thermodynamic stability usually requires the formation enthalpy and cohesive energy to be negative.This work provides the calculated formation enthalpies of −0.151, −0.17, −0.Therefore, the most thermodynamically stable structure refers to Al 3 Zr phase regardless of its polymorphs, while Al 5 Cu 2 Mg 8 Si 6 behaves the value of −0.02 eV (close to zero) indicating its metastable nature, which may decompose after high-temperature artificial aging.
The second order elastic constants were calculated and listed in Table 2, and the corresponding polycrystalline bulk and shear moduli are given in Table 3 calculated by Reuss and Voigt methods [38], respectively; the real polycrystalline values are estimated by Hill's average [39].For Al 2 Cu and Al 2 CuMg, C 11 , C 22 , and C 33 are over 100 GPa, which are much higher than other elastic constants, indicating their strong anti-compressibility along principal axes.Especially, Al 2 Cu shows the same stiffness along the aand baxes, both of which are weaker than that along the caxis, while Al 2 CuMg is stiffer along the baxis compared with the aor caxis.The tabulated elastic modulus (C 44 ) of both Al 2 Cu and Al 2 CuMg is significantly small.This corresponds to the shear mode and indicates their relatively small shear modulus (~43 GPa as shown in Table 3).Al 3 Fe, Al 3 Fe 2 Si, and Al 7 Cu 2 Fe show much higher elastic constants of C 11 , C 22 , and C 33 (over 200 GPa), implying these phases are very hard to be deformed.Al 5 Cu 2 Mg 8 Si 6 is an exception; all elastic constants are much smaller than others, implying its soft nature.Actually, Al 5 Cu 2 Mg 8 Si 6 is a mechanically unstable phase since it disobeys the well-known Born-Huang stability criterion [40,41].For Mn-based phases Al 6 Mn and Al 20 Cu 2 Mn 3 , Al 6 Mn shows better anticompressibility along the principal axis than Al 20 Cu 2 Mn 3 , while the shear or complex mode corresponding to C 44 or C 12 and C 23 of Al 6 Mn are much smaller than that of Al 20 Cu 2 Mn 3 .Al 3 Zr usually has three kinds of polymorphs; the calculated results of both D0 22 and L1 2 phases are very similar to other references as listed in Table 1, while for the D0 23 phase, the reference calculation results are dispersed from 201 to 284 GPa by taking C 11 as an example [42].To our knowledge, most precipitates of the 2xxx series aluminum alloys have not any experimental values, our study may provide valuable data especially for Al 3 Fe 2 Si, Al 6 Mn, etc.Based on the values of the Voigt-Reuss-Hill approximation shown in Table 2, the Young's modulus and Poisson's ratio can be calculated by E = 9BG/(3B + G) and σ = (3B − 2G)/(6B + 2G), respectively.Generally, the mechanical moduli of precipitates of 2xxx series aluminum alloys are similar to previously reported values.As shown in Figure 2, the variation trend of bulk, shear, and Young's muduli shares a similar tendency, in which Al 3 Fe 2 Si behaves the highest values.Bulk modulus, for instance, which is derived from 63.5 to 155.9 GPa, has the trend of: Al  [47]).Young's modulus usually reflects the plastic deformation ability of bulk materials; this work proves the ultrahigh anti-deformation nature of Al 7 Cu 2 Fe.For three polymorphs of Al 3 Zr, the Young's modulus of D0 23 and D0 22 phases are similar (~196 GPa), which are much higher than the L1 2 phase (~156 GPa), indicating their outstanding mechanical behavior.Typically, for covalent and ionic materials, the value of Poisson's ratio is 0.1 and 0.25, respectively, whereas for metallic materials, the value is 0.3 [48].Most of second phases have the Poisson's ratio of ~0.a, b, c, d, e, f, g, h, i, j, k, l, m, and n are the theoretical data form Refs. [22][23][24][25][26][27][28]43,44,[50][51][52], and [45], respectively.
The mechanical anisotropy of crystals is also characterized by using 3-D mechanical modulus and anisotropy indexes.Firstly, the directional dependence of the Young's modulus for different crystals can be evaluated by [  The mechanical anisotropy of crystals is also characterized by using 3-D mech modulus and anisotropy indexes.Firstly, the directional dependence of the Young's ulus for different crystals can be evaluated by [53]: For cubic (Al 3 Fe 2 Si, Al 3 Zr_L1 2 ): For monoclinic (Al 5 Cu 2 Mg 8 Si 6 ): In the above equations, l 1 = sin θ cos ϕ, l 2 = sin θ sin ϕ, and l 3 = cos θ.The results are shown in Figure 3. Al 2 Cu and Al 5 Cu 2 Mg 8 Si 6 show significantly stronger anisotropy than other phases, since the surface contours show a large deviation from the perfect spherical shape, while the iron-based compounds Al 3 Fe 2 Si and Al 7 Cu 2 Fe are very isotropy.As for the three Al 3 Zr phases, the L1 2 structure shows the best isotropic nature.More direct information can be seen from the planar projections on the (001) and (110) crystal planes in Figure 4. On the (001) plane, Al 2 CrMg shows the strongest anisotropy character, while on the (110) plane, Al 5 Cu 2 M 8 Si 6 has the extremely strong anisotropy as can be seen from the green line in Figure 4b.Three commonly used anisotropy indexes (A B , A G and A U ) were calculated and shown in Table 3, where   Three commonly used anisotropy indexes (AB, AG and A U ) were calculated and shown in Table 3, where AB = (BV − BR)/(BV + BR), AG = (GV − GR)/(GV + GR), and A U = 5(GV/GR) + (BV/BR) − 6.For Al2Cu and Al2CuMg [54], the calculated AB of Al2Cu is zero, which indicates the perfect isotropy of Al2Cu in compression, while Al2CuMg has a high value of AB indicating its anisotropic nature.As for shear mode, Al2Cu has a high value of AG, implying its anisotropic character.The universal anisotropy index of A U follows the trend of:

Thermophysical Properties
The precipitation influences thermophysical properties such as thermoelectric power, the thermal expansion coefficient, and thermal conductivity.In this part, the sound velocity and the Debye temperature were calculated.The longitudinal, transverse, and average wave velocities are given by [55]:

Thermophysical Properties
The precipitation influences thermophysical properties such as thermoelectric power, the thermal expansion coefficient, and thermal conductivity.In this part, the sound velocity and the Debye temperature were calculated.The longitudinal, transverse, and average wave velocities are given by [55]: Then, the Debye temperature is given by [56]: where Θ D is the Debye temperature, and h, k B and N A are the Planck, Boltzmann and Avogadro constants, respectively; n is atomic number; M is molecular weight, and ρ is volumetric density; v l , v t and v m are longitudinal, transverse, and average acoustic velocities, respectively.The results are shown in The thermal expansion is an important thermophysical character, which characterizes the anharmonicity of crystals.At different temperatures, the Helmholtz free energy of Al 2 Cu is shown in Figure 5a, based on which the volume expansion (∆V/V 0 ) and bulk modulus can be fitted by the well-known Birch-Murnaghan's equation of states (EOS) [57].Figure 5b shows the volume' expansions (∆V/V 0 ) versus temperature curve by taking Al 2 Cu as an example.Then, the linear thermal expansion coefficient (LTEC) is obtained and plotted in Figure 6.For the Al 2 Cu phase, the LTEC along the a or c axis is also calculated, which implies that the value of α c is much higher than that of α a .The calculated average LTEC is ~16.2 ppm K −1 from 300 to 800 K, which is similar to the experimental data (~17.2ppm K −1 ) [33].For Al 3 Zr polymorphs, the results are plotted in Figure 6b, and our results indicate that the L1 2 phase has the highest LTEC value, followed by D0 23 , and the lowest one refers to D0 22 .From 300 K to 800 K, our calculated values are in extreme agreement with the calculated data given by Saha et al. [58], although both of our curves are slightly higher than experimental curve [59], as shown in Figure 6b.The calculated LTEC of three Al 3 Zr polymorphs follow the order of L1 2 > D0 23 > D0 22 ; all of them show a much lower value than pure Al substance.Overall, all calculated average LTECs are given in Figure 6c; in this figure the pure Al substance is also plotted.Al 2 CuMg shows the highest LTEC, followed by Al 3 Fe, Al 2 Cu, Al 3 Zr_L1 2 and others, while Al 3 Zr_D0 22 is the lowest one; the discrepancy between a-Al and Al 2 CuMg is the smallest, which may decrease the heat misfit degree between them and improve the thermal shock resistant property, and thereby delay the initiation and propagation of thermal crack at the interface.in Figure 6c; in this figure the pure Al substance is also plotted.Al2CuMg shows the highest LTEC, followed by Al3Fe, Al2Cu, Al3Zr_L12 and others, while Al3Zr_D022 is the lowest one; the discrepancy between a-Al and Al2CuMg is the smallest, which may decrease the heat misfit degree between them and improve the thermal shock resistant property, and thereby delay the initiation and propagation of thermal crack at the interface.in Figure 6c; in this figure the pure Al substance is also plotted.Al2CuMg shows the highest LTEC, followed by Al3Fe, Al2Cu, Al3Zr_L12 and others, while Al3Zr_D022 is the lowest one; the discrepancy between a-Al and Al2CuMg is the smallest, which may decrease the heat misfit degree between them and improve the thermal shock resistant property, and thereby delay the initiation and propagation of thermal crack at the interface.The specific heats at constant pressure (CP) and constant volume (CV) are two fundamental parameters, which can be related by [62]: CP−CV = λ 2 V(T)TB (10) where λ refers to the volumetric TEC, V(T) and B are the volume and bulk modulus at temperature T. The results are plotted in Figure 7; Al3Cu2Fe has much higher heat capacity than other phases.At very low temperatures, all curves increase rapidly due to the crystalline lattice vibration, and then the increasing rate reduces slowly; for CV, the curves tend to a well-known limit of Dulong-Petit, while for CP the curves keep increasing due to the work done by lattice expansion.The specific heats at constant pressure (C P ) and constant volume (C V ) are two fundamental parameters, which can be related by [62]: where λ refers to the volumetric TEC, V(T) and B are the volume and bulk modulus at temperature T. The results are plotted in Figure 7; Al 3 Cu 2 Fe has much higher heat capacity than other phases.At very low temperatures, all curves increase rapidly due to the crystalline lattice vibration, and then the increasing rate reduces slowly; for C V , the curves tend to a well-known limit of Dulong-Petit, while for C P the curves keep increasing due to the work done by lattice expansion.
Based on the curves of (F (V, T) − V), the bulk modulus and its pressure derivative at different temperatures can be fitted based on Birch-Murnaghan EOS.The anticompressibility of the second phases of 2xxx series aluminum alloys are given in Figure 8.At 0 K, Al 3 Fe, Al 3 Fe 2 Si, and Al 7 Cu 2 Fe are hard and anti-compressive, while Al 2 CuMg is the softest phase; meanwhile, three Al 3 Zr phases locate between them, which show moderate anti-compressibility.At temperatures of 300 or 700 K, the overall sequences of second phases are similar; Al 2 CuMg shows the most compressive character.However, with the increase of temperature, Al 2 Cu becomes more and more soft, and its compressibility tends to be similar to the Al 2 CuMg phase at 700 K. Based on the curves of (F (V, T) − V), the bulk modulus and its p different temperatures can be fitted based on Birch-Murnaghan EOS ibility of the second phases of 2xxx series aluminum alloys are give

Metals 2022 , 18 Figure 1 .
Figure 1.The unit crystal cells of second phases in the 2xxx series aluminum alloys.

Figure 1 .
Figure 1.The unit crystal cells of second phases in the 2xxx series aluminum alloys.

Figure 2 .
Figure 2. Mechanical properties of second phases in the 2xxx series aluminum alloys: (a) bulk and Young's moduli, (b) Vicker's hardness, and (c) Poisson's ratio, and brittle index; the hor lines refer to the typical Poisson's ratio value (~0.3) for pure metal, and criterion of ductility/ ness (B/G = 1.75).

Figure 2 .
Figure 2. Mechanical properties of second phases in the 2xxx series aluminum alloys: (a) bulk, shear, and Young's moduli, (b) Vicker's hardness, and (c) Poisson's ratio, and brittle index; the horizontal lines refer to the typical Poisson's ratio value (~0.3) for pure metal, and criterion of ductility/brittleness (B/G = 1.75).

18 Figure 3 .
Figure 3.The surface constructions of the Young's modulus of second phases in the 2xxx series aluminum alloys.

Figure 3 .
Figure 3.The surface constructions of the Young's modulus of second phases in the 2xxx series aluminum alloys.

For Al 2
Cu and Al 2 CuMg [54], the calculated A B of Al 2 Cu is zero, which indicates the perfect isotropy of Al 2 Cu in compression, while Al 2 CuMg has a high value of A B indicating its anisotropic nature.As for shear mode, Al 2 Cu has a high value of A G , implying its anisotropic character.The universal anisotropy index of A U follows the trend of: Al 2 Cu > Al 2 CuMg ≈ Al 3 Zr_D0 22 ≈ Al 20 Cu 2 Mn 3 > Al 3 Fe ≈ Al 6 Mn > Al 3 Zr_D0 23 ≈ Al 3 Zr_ L1 2 > Al 7 Cu 2 Fe > Al 3 Fe 2 Si.

Figure 3 .
Figure 3.The surface constructions of the Young's modulus of second phases in the 2xxx series aluminum alloys.

Figure 4 .
Figure 4.The planar projections of the Young's modulus of second phases in the 2xxx series aluminum alloys: (a) (001) crystal plane, and (b) (110) crystal plane.

Figure 4 .
Figure 4.The planar projections of the Young's modulus of second phases in the 2xxx series aluminum alloys: (a) (001) crystal plane, and (b) (110) crystal plane.

Figure 5 .
Figure 5. Dependence of the Helmholtz free energy, F (V, T), on the crystal volume under different temperatures (a) by taking Al2Cu for example, based on which the volume' expansions (ΔV/V0) are given in (b).

Figure 5 .
Figure 5. Dependence of the Helmholtz free energy, F (V, T), on the crystal volume under different temperatures (a) by taking Al 2 Cu for example, based on which the volume' expansions (∆V/V 0 ) are given in (b).

Figure 5 .
Figure 5. Dependence of the Helmholtz free energy, F (V, T), on the crystal volume under different temperatures (a) by taking Al2Cu for example, based on which the volume' expansions (ΔV/V0) are given in (b).

Figure 6 .
Figure 6.Linear thermal expansion coefficients (LTECs) of second phases in the 2xxx aluminum alloys: (a) LTEC along a or c principal axis, and its average by taking Al2Cu for example, the references [33,60,61] are also ploted in this sub-figure; (b) average LTEC of Al3Zr by considering different polymorphs with references [58,59]; (c) LTEC of all second phases versus temperature, and calculated results of pure Al from reference [58].

Figure 6 .
Figure 6.Linear thermal expansion coefficients (LTECs) of second phases in the 2xxx aluminum alloys: (a) LTEC along a or c principal axis, and its average by taking Al 2 Cu for example, the references [33,60,61] are also ploted in this sub-figure; (b) average LTEC of Al 3 Zr by considering different polymorphs with references [58,59]; (c) LTEC of all second phases versus temperature, and calculated results of pure Al from reference [58].

sFigure 7 .
Figure 7.The specific heats at constant volume (a) or pressure (b) of sec aluminum alloys.

Figure 7 .Figure 8 .
Figure 7.The specific heats at constant volume (a) or pressure (b) of second phases in the 2xxx aluminum alloys.

Table 2 .
Theoretically calculated elastic constants (C ij in GPa) for second phases of 2xxx series aluminum alloys.

C 11 C 12 C 13 C 22 C 23 C 33 C 44 C 66
3 Fe 2 Si > Al 3 Fe > Al 7 Cu 2 Fe > Al 6 Mn ≈ Al 3 Zr ≈ Al 20 Cu 2 Mn 3 > Al 2 Cu > Al 2 CuMg > Al 5 Cu 2 Mg 8 Si 6 .All iron-based compounds of Al 3 Fe 2 Si, Al 3 Fe, and Al 7 Cu 2 Fe have a large bulk modulus, which is more or less smaller than BCC-iron (~174 GPa 3, such as Al 2 Cu, Al 3 Fe, Al 5 Cu 2 Mg 8 Si 6 , and Al 20 Cu 2 Mn 3 , indicating their advanced metallic nature.For Al 2 CuMg, Al 3 Fe 2 Si, Al 6 Mn, and Al 3 Zr, they are dominated by a mainly covalent bond.The brittle index of B/G is applied to analyze the ductility of phases.The higher the value of B/G, the better the ductility of the materials.The present work indicates that Al 2 Cu, Al 3 Fe, Al 5 Cu 2 Mg 8 Si 6 , and Al 20 Cu 2 Mn 3 are ductile phases, which are in agreement with their advanced metallic nature.For the three Al 3 Zr polymorphs, the L1 2 phase shows the best ductility.Furthermore, in the present work, a semi-empirical model proposed by Chen et al. [49] was employed to evaluate the Vicker's hardness.As shown in Table 3, the hardness of Al 3 Zr_ D0 22 and D0 23 phases is very high (~18GPa), which is comparable to the value of Al 7 Cu 2 Fe; whereas Al 5 Cu 2 Mg 8 Si 6 is quite soft since its hardness is only 2.7 GPa.The common precipitates of Al 2 Cu and Al 2 CuMg show a moderate hardness of 4.3 GPa and 6.8 GPa, which are much softer than Al 6 Mn or Al 3 Zr phases.

Table 3 .
Theoretically calculated elastic properties including bulk modulus (B in GPa) and its pressure derivative (B ), shear modulus (G in GPa), Young's modulus (E in GPa), Poisson ratio (σ), and anisotropy factors (A B , A G and A U ) for second phases of 2xxx series aluminum alloys.

Table 4 ,
the sound velocity of Al 3 Fe 2 Si is highest among all considered phases, since it behaves the greatest mechanical moduli and small density, and therefore, the Debye temperature of Al 3 Fe 2 Si is as high as 622 K.The Debye temperatures obey the trend of: Al 3 Fe 2 Si > Al 3 Zr_D0 23 > Al 6 Mn > Al 3 Zr_D0 22 > Al 7 Cu 2 Fe > Al 3 Fe > Al 2 CuMg > Al 20 Cu 2 Mn 3 > Al 2 Cu > Al 3 Zr_L1 2 .

Table 4 .
Calculated sound velocities (km/s) of second phases of 2xxx series aluminum alloys; the Debye temperatures (K) are also shown below.